Did you read the thread? Your question stated that photons could be described by probability waves and then asked a question that if that were were the case...... The lack of responses would suggest that it cannot be the case.

Did you read the thread? Your question stated that photons could be described by probability waves and then asked a question that if that were were the case...... The lack of responses would suggest that it cannot be the case.

A photon is not a non-relativistic object, this implies that its quantum behavior cannot be described by the (non-relativistic) Schroedinger equation. For non-relativistic particles, you can find its position representation wavefunction by solving the Schroedinger equation. Since you can't do that with photon, there is no spatial wavefunction in the usual sense of non-relativistic QM which can be associated to photons.

what is the relationship between the electric field and the probability amplitude?

Photon state is described by the so-called Fock states or its linear combination. Whereas the electric field, since it's a physical quantity, is associated with a field operator ##\tilde{E}## and the electric field you observe is the average value of this operator with respect to the particular state of the photon, i.e. ##\langle \psi | \tilde{E} | \psi\rangle##. The state ##|\psi\rangle ## can take a number of forms, each of which is a particular linear combination of Fock states, for example number state, coherent state, squeezed state, etc, and again they do not have position representation.